\subsection{Bogliubov quasiparticle for Fermionic excitation}
I can work out the Fermionic excitation by first linearize the (reduced pairing) hamiltonian and then use the canonical transformation\cite{Tinkham}.  The coefficients of the transformation is messy, but the linearized hamiltonian provides the $3\times3$ matrix of the $(a^{\dagger}_{\vk},\; b^{}_{-\vk},\;c^{}_{-\vk})$  and their hermitian conjugate.  The three eigenvalues give the fermionic single pair excitation and eigenvectors give the exact coefficient of transformation.  The secular equation is a cubic equation.  In principle, it has analytical roots, but they are very nasty and do not generate much intuition.  Instead a zeroth order approximation where no Pauli exclusion of two-channel is used and then correction is sought.  

With this perturbative approach, it is a bit difficult to follow \cite{Rickayzen}'s way to canonical transform first and then find equation of motion of collective mode.  

One crucial assumption is made that the close-channel bound state is small, it is small comparing to average inter-particle distance.  This assumption is justified for the category of the problem.  Otherwise, it is no longer a clear Feshbach resonance problem but a problem with two interacted degenerate fermi gas (both in superfluid state possibly? ).  